Traveling Waves

To simplify things, we will set the phase angle equal to zero and rewrite this equation as

y(x,t) = A sin (k (x + (ω/k)t)).

Compare this to the form of a function which undergoes translation: the transformation

y(x) -> y(x + a)

moves the function to the left a units:

This shows us that our traveling wave is simply an ordinary sine wave which is being translated by an amount depending on time. As we saw in the last section, w/k is the speed of the wave which in this case is traveling to the left:

Each oscillator generates a hemispherical wave disturbance in the direction of travel, which we have shown here in cross section as semicircles. The wave front is tangent to the semicircles at any given time.

Recall the wave demonstration applet in the previous section. Set the black and blue waves to the same mode number and observe the red superposition wave as you vary the blue wave phase angle. When the phase angle is zero, the black and blue waves are said to be in phase and constructively interfere. When the phase angle is π or -π, the superposition wave has an amplitude of zero and the black and blue waves are said to be out of phase: they destructively interfere. The interference of coherent waves (which start out in phase) lies at the heart of the quintessential wave phenomenon: diffraction.

We start by illustrating how interference works in two dimensions with this applet, which shows the superposition of two waves from
sources at the bottom of the image. The white spots are where the waves
constructively superpose and are in phase, and the black spots are where the waves destructively superpose and are
out of phase:

Suppose that a beam of light is incident on a surface with two slits in it (separated by a distance d). Huygens' principle states that the two slits will act as coherent sources. These correspond to the applet above, with equal frequencies and zero phase angle. When the light rays from these two sources interfere at the screen (a distance L away from the slits), one will have traveled farther than the other:

The angles are approximately equal, as are the hypotenuse and adjacent sides of the large triangle, if the angle is small. This is true since the length L is usually much greater than the distance between the slits. The difference in length of the two rays from the slits to their point of intersection is d sin θ. When this length difference is an integral number of wavelengths:

d sin θ = m λ,

the phase difference is

k Δx = (2 π / λ) * m λ

= 2 π m,

so the waves are in phase, and we have constructive interference and a bright spot at the point of intersection (at x = L sin θ). When this length difference is a wavelength times an integer plus one half:

d sin θ = (m + 1/2) λ,

the phase difference is

k Δx = (2 π / λ) * (m + 1/2) λ

= 2 π m + π,

so the waves are π radians out of phase, and we have destructive interference with a dark spot at the intersection. If we label the minima and maxima with
a number "a" as follows:

we can write their locations as

x = a L λ / d.

The first applet plots the intensity as a function of position, while the second applet provides a visual illustration of the actual diffraction pattern: